There are several different types of engines. The most common types are 4-stroke and 2-stroke engines. "Cycle" and "Stroke" are interchangeable terms in this context -- it means the same thing.
All modern car engines are of the 4-stroke type. Some smaller engines (leaf blowers, generators, lawnmowers, and some motorcycles) are 2-stroke.
2-stroke engines use a mixture of fuel and oil, which is used to power the engine and provide lubrication simultaneously. These engines have a characteristic smell and a bluish tint to the exhaust. 2-cycle oil is meant to be mixed with the fuel (usually at a ratio of around 30 parts fuel to 1 part oil), and that mixture is poured into the fuel tank, which will usually have a "Fuel+Oil" label on the cap.
4-stroke engines keep fuel and oil separate at all times. Oil is usually added at a filler cap on the valve cover (often labeled with the manufacturer's recommended viscosity, like 5W30 or 10W40), where it drains through the engine to an oil pan at the bottom, and is then picked up by an oil pump, passed through a filter, and pumped back up to various points in the engine.
2-stroke engines are smaller and lighter than 4-stroke engines that generate equivalent power, so they are useful in handheld equipment like leaf blowers. They are ideal in situations where size and weight are critical, and where the engine will be run at a constant high RPM for long periods of time.
4-stroke engines are larger and heavier, but offer cleaner emissions and a wider power range. They are better suited to situations where the engine will be run at low or varying RPMs.
One easy way to determine what kind of engine you're working with is that 2-stroke engines don't have oil filters or oil dipsticks.
"Premium" doesn't mean anything specific as it applies to motor oil of either type, but a premium oil may in fact be of higher quality.